1. Introduction

The use of satellite altimeter data to study the circulation of the Mediterranean Sea is complicated by a poor signal-to-noise ratio (Larnicol et al. 1995): the sea level anomaly signal is at most 10 cm rms in the energetic regions of the Mediterranean, while the residual altimeter noise, including orbit and atmospheric load errors, is of the order of 5 cm rms or more. By construction, the sea level anomaly signal, which is here obtained by removing a 4-yr temporal mean at every point, does not contain any geoid errors, except for those due to lateral orbit errors over steep topography. The major sources of low wavenumber uncertainty are the orbit errors (Marshall et al. 1995) and, in the Mediterranean, the atmospheric loading not removed by the inverse barometer correction (Candela and Lozano 1994; Le Traon and Gauzelin 1997). Orbit and atmospheric loading errors are removed in practice, along with some of the oceanic signal, by fitting and subtracting low-order polynomials or cubic splines from each altimeter track so as to minimize track-to-track crossover differences (Le Traon et al. 1995b; Larnicol et al. 1995; Bonnefond et al. 1995; Vasquez-Cuervo et al. 1996). The problem with these methods is that they do not make use of an explicit statistical model for the low wavenumber errors: the crossover difference minimization algorithms are ad hoc, and the estimation of sea surface topography and of orbit/atmospheric load corrections are treated as two separate steps, that is, the uncertainties in the orbit correction step are not taken into account when estimating the sea surface topography.

In this paper, we use a multiresolution optimal interpolation scheme to map the sea level anomaly of the Mediterranean Sea based on TOPEX/Poseidon and ERS-1 data. The departure of our work from previous efforts is that we jointly estimate the sea level anomaly and the orbit/atmospheric load errors, so that the uncertainties in the errors are naturally reflected in the estimates and vice versa. This approach requires explicit statistical models for the sea level anomaly and for the orbit and atmospheric load errors, and in turn provides estimation error variances for all estimates at multiple scales. The major advantage of the multiresolution scheme is that its numerical implementation is extremely efficient, and therefore the estimation can be repeated a large number of times to test different statistical models and to obtain consistent results.

The multiresolution optimal interpolation scheme has previously been described and used for mapping altimeter data in the North Pacific Ocean by Fieguth et al. (1995), but here it is substantially modified to allow for the joint estimation of orbit and atmospheric loading errors. The scheme is based on a scale-recursive statistical description of the processes under study. This particular statistical description permits the use of recursive estimation algorithms that provide a significant speed advantage over more conventional estimation algorithms. One difficulty associated with the implementation of such multiresolution optimal interpolation schemes is in the construction and specification of the multiscale statistical models. For this reason we are making our algorithm available on the Internet in the form of a Matlab-callable routine (see the appendix). In a related paper, Menemenlis et al. (1997a) describe a multiscale statistical model that can be applied to a more general class of estimation problems (again, with an algorithm available on the Internet).

2. Statistical modeling

An abstract statistical description of the TOPEX/Poseidon or ERS-1 altimeter data may be written as follows:

View Expanded

where ζ(x, y, t) represents the oceanographic sea level anomaly signal taken at time t and location (x, y), and h(t) represents the altimetric measurement of ζ(x, y, t) subject to noise υ(t), which includes residual tidal, atmospheric, and orbit errors. The sea surface height ζ and the measurement noise υ are modeled to be zero mean (i.e., systematic signal components are assumed to have been removed), so that 〈υ(t)〉 = 0, 〈ζ(x, y, t)〉 = 0. Here R(t, t) and P(x, x, y, y, t, t) represent the second-order prior statistics of υ(t) and ζ(x, y, t), respectively. Solving for the time-varying process ζ(x, y, t) based on the prior statistics and measurements in (1)–(3) is extremely difficult, predominantly due to the complicated ocean surface dynamics captured by P(x, x, y, y, t, t). Instead, for the purposes of this paper, we consider the ocean surface to be static (a reasonable assumption for the large-scale, baroclinic modes of variability over periods up to 10 days in length) and statistically stationary;

ζx, y, tζxytPxxyy(4)

Motivated by spectral analyses of the altimetric data and earlier work (Wunsch and Stammer 1995; Fieguth et al. 1995; Gaspar and Wunsch 1989), we propose to model the surface ζ(x, y, t) as a 1/k^μ process, where k is the horizontal wavenumber.

Studies (Fu et al. 1994; Le Traon et al. 1994; Le Traon et al. 1995b; Le Traon et al. 1997) of the measurement noise suggest that the TOPEX/Poseidon or ERS-1 noise process υ(t) can be modeled as the sum of two separate random processes:

υtυ_WGNtυ_CORRt(5)

On the one hand, the random measurement error of each altimeter has a very short correlation period and is here modeled by a Gaussian white noise process υ_WGN(t). On the other hand, the uncertainties in the orbital position of the satellite and in the atmospheric load correction are a correlated process of low wavenumber, represented here by υ_CORR(t). Earlier altimetric optimal interpolation efforts have either ignored υ_CORR (Fieguth et al. 1995) or tried to remove it via a preprocessing step (Larnicol et al. 1995)—an approach that requires ad hoc assumptions and that fails to take into account the posterior error statistics of the preprocessing step when estimating ζ(x, y, t). Instead, we propose to use an estimation framework that is flexible enough to allow us to specify the noise model (5) explicitly. For relatively small geographic areas such as the Mediterranean, the low-wavenumber error may be modeled as a low-order polynomial, leading to the following noise model:

View Expanded

where J represents the order of the polynomial correction and α_j(i) represents the unknown jth order polynomial coefficient for satellite track i. We model the α_j(i) as being uncorrelated with each other and with υ_WGN; that is, each track is modeled as having independent errors. Although the estimator is capable of polynomial corrections to an arbitrary degree, we have chosen to apply first order, J = 1 (i.e., to estimate bias and tilt) corrections only, which is consistent with the low-order corrections applied by previous authors [Larnicol et al. (1995) used J = 0, Vasquez-Cuervo et al. (1996) used J = 1] so as to minimize the attenuation of the low-wavenumber oceanographic signal. The next step is to specify prior statistics for (6) in the form of 〈υ_WGN(t)²〉 and 〈α_j(i)²〉, and this completes the statistical description of the estimation problem.

In principle, the optimal interpolation (minimum variance) solution for the above estimation problem could be obtained by matrix inversion. However, the fields computed in section 3 are 100 × 225 pixels in size, and directly estimating such fields would require the inversion of a 22500 × 22500 matrix. In practice, such an approach is computationally infeasible, so we propose to use a recently developed multiscale estimation framework (Chou et al. 1994; Luettgen et al. 1994; Fieguth et al. 1995), which is capable of solving very large estimation problems with a modest amount of computational effort. Specifically, this framework will be capable of jointly computing the sea surface and the orbit/atmospheric load errors, while still computing estimation error statistics. This computational advantage does not come for free: It can be a significant challenge to develop multiscale models that capture the desired statistics and the remainder of this section will address this challenge. In particular, the correlated noise term in (5) and (6), which was ignored in the model of Fieguth et al. (1995), required a nontrivial effort to model.

The ocean surface of interest is modeled as the finest scale of a multiscale stochastic process z(s) defined on a quad-tree, that is, a tree on which each node has four descendents (except those nodes on the finest scale). The process z(s) and its associated measurement process h(s) are modeled as

View Expanded

where s indexes the nodes of the multiscale tree with s = 0 indicating the root node of the tree, sγ represents the parent of node s, and w(s) is a white noise process;for our altimetric problem, measurements exist only at the finest scale of the tree. The multiscale state z(s) is made up of bias and tilt parameters for each track, along with a multiresolution process ζ(s), which represents the ocean surface ζ(x, y, t);

zsζsα₀α₁α₀M₁M^T(8)

where M counts the number of satellite tracks present in the measurements and where we have assumed that J = 1, that is, a first-order polynomial correction for the low-wavenumber errors. As in Fieguth et al. (1995), ζ(s) is a multiresolution description of the sea surface height: On finescales of the tree, ζ(s) captures the local sea surface height and on coarse scales ζ(s) captures broad averages. [For reasons of efficiency our software uses a reduced order z(s). In particular, only those α_j(i) corresponding to tracks that are measured by the children of s need to be kept in z(s). However, for clarity we shall retain the state definition of (8) throughout this paper.]

Given this definition of the state, (8), three quantities must be specified to complete the multiscale model: 1) a prior covariance P(0) at the root node of the tree consistent with 〈α_j(i)²〉; 2) a multiscale state model, A(s) and B(s), consistent with (8) and the 1/k^μ prior on ζ(s);and 3) a multiscale measurement model, C(s) and R(s), consistent with (1), (6), and 〈υ_WGN(t)²〉.

The model for z(s) due to the α_j(i) is simple: the coefficients α_j(i) are just copied unchanged from parent to child. For the sea surface ζ(s) we use the 1/k^μ multiscale model of Fieguth et al. (1995): the scale-to-scale variations in ζ(s) are random with exponentially decreasing variance. The resulting multiscale model is

View Expanded

where m(s) denotes the scale of node s on the multiscale tree. A discussion regarding the particular choice of B_o, μ, and 〈ζ(0)²〉 may be found in Fieguth et al. (1995).

Finally, we need to define multiscale measurements at the finest scale of the tree, consistent with (1) and (6);

View Expanded

where N is the total number of tracks, i₁, . . . , i_N, which pass through the finest-scale pixel corresponding to node s. The N measurements are taken at times t₁, . . . , t_N, and l(s) represents the latitude in degrees at which these measurements were taken. [The latitude l(s) relative to 37°N is used as a convenient surrogate for time t in (6);this replacement is possible because along each track the measurement latitude is a nearly linear function of the measurement time.] Furthermore,

Rsυ_WGNt₁²υ_WGNt_N²(13)

where the value assumed by each 〈υ_WGN(t)²〉 along the diagonal of R(s) depends upon the satellite that took the measurement. At this point the description of the multiscale model structure is complete, and it is this model that will be used to compute the experimental results in the next section.

3. Experimental results

To illustrate the application of the multiresolution estimator described above, we have mapped sea level anomaly TOPEX/Poseidon and ERS-1 data in the Mediterranean. TOPEX/Poseidon data were processed and interpolated onto a fixed 6-km equidistant along-track grid by King et al. (1994), which includes the standard environmental, inverse barometer, and orbit error corrections. Tidal corrections are from the CSR3.0 (Shum et al. 1997) model. The ERS-1 data are from the files distributed by AVISO (Le Traon et al. 1995a). In addition to the standard environmental corrections, the AVISO data include corrections of the ERS-1 orbit errors based on a global minimization of TOPEX/Poseidon–ERS-1 crossover differences (Le Traon et al. 1995b). [This ad hoc orbit correction is not necessary since the multiresolution estimation framework can accommodate different bias and tilt priors, 〈α_j(i)²〉, for each altimeter track. Nevertheless, sufficient orbit/atmospheric load errors remain to demonstrate the power of the multiresolution estimator.] A 4-yr (1 yr for the ERS-1 altimeter) mean sea level is computed and subtracted from the data to obtain sea level anomaly.

The following sections compare and contrast sea level anomaly maps of the Mediterranean Sea produced using several different methods: 1) simple interpolation, 2) multiscale estimation without polynomial corrections as in (Fieguth et al. 1995), 3) multiscale estimation with first-order polynomial corrections as discussed in section 2, and 4) a suboptimal space–time optimal interpolation algorithm. Each map is computed from the measurements of one 10-day TOPEX/Poseidon repeat cycle; in particular, we used TOPEX data from repeat cycle 24 (9–19 May 1993), which exhibited significant correlated measurement errors and visually demonstrates the removal of residual orbit/atmospheric load errors. Finally, we show results from a joint TOPEX/ERS-1 estimation.

In each case, the anomaly map is a grid of 100 × 225 estimates. The multiscale estimates are computed on a tree having nine scales, which corresponds to a finest scale of 256 × 256 pixels in size; each pixel is square, 0.16 degrees on a side. Our algorithm requires that the measurements be defined on the finest scale of the multiscale tree and therefore the data need to be repositioned by up to ±1/2 pixel width. For this reason the pixel size must be chosen sufficiently fine for this repositioning to be trivial. The associated map is then just a particular 100 × 225 pixel subset of the finest scale. Prior statistics were obtained empirically from TOPEX/Poseidon repeat cycles 3 to 161 (October 1992–February 1997) and from ERS-1 repeat cycles 6 to 18 (October 1992–January 1994).

4. Concluding remarks

Most of our past efforts (Chou et al. 1994; Luettgen et al. 1994; Fieguth et al. 1995) with respect to the multiscale framework have concentrated on developing models and on understanding the statistical properties of the framework. We now believe that our understanding has improved to the point where our framework can make real scientific contributions and be used by other researchers as a tool for solving related estimation problems (Menemenlis et al. 1997a; Menemenlis et al. 1997b).

The particular mapping example discussed in this paper assumes a model on a square grid that is stationary, 1/k-like, and static in time. The assumption of a stationary model on a square grid admittedly restricts this software to regional investigations since at ocean basin scales the curvature of the earth is inconsistent with such a model. The use of a 1/f-like prior model leads to a simple model in our multiscale framework; other prior models are possible (Menemenlis et al. 1997a) but require more complicated multiscale implementations. Finally, the assumption that the ocean is static limits the time window of measurements to be used in producing a single map. However, the following three aspects of the multiscale method lend themselves to processing dynamic data.

1. The computational efficiency of the framework makes it practical to compute large numbers of maps, successive in time, which could then be interpolated.

2. The present framework’s tolerance for nonstationarities allows the definition of a time-varying measurement noise variance. For example, (13) can be modified such that the diagonal elements of R(s) take the form

   υ_WGNt_i²ϕt_it_o(14)

   where ϕ(|t_i − t_o|) is a positive, monotonically increasing function and t_o is a given time origin. Such a definition of R(s) would have the effect of downweighting measurements that lie away from the time origin t_o (Fieguth 1995).

3. In principle an oct-tree could be constructed—multiscale in two space dimensions and one in time. This would be a very elegant solution to the nonstationarity issue; however, the development of efficient models on such trees presents formidable challenges and is the subject of ongoing research.

It must be pointed out, however, that the regional and time-stationary limitations are inherent in our particular chosen model, not in the multiscale framework itself. The development of nonstationary (Fieguth et al. 1995; Fieguth 1995) or temporally dynamic multiscale models is also the subject of ongoing research.